BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

The Bonhöffer-van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions are disappeared simultaneously by the homoclinic bifurcation on the Poincaré map and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it is disappeared by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments

A DESIGN METHOD OF BURSTING USING TWO-PARAMETER BIFURCATION DIAGRAMS IN FITZHUGH–NAGUMO MODEL

Spiking and bursting observed in nerve membranes seem to be important when we investigate information representation model in the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose a design method to generate bursting responses in FitzHugh-Nagumo model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the 2-parameter bifurcation diagram

シテイ シタ ヘンカク オ ユウスル コテイテン ノ ケイサン ト ソノ オウヨウ

非線形力学系において，系に含まれるパラメータが変化し，固定点の特性乗数が複素単位円外に出ることにより各種の局所的分岐現象を生じる．本報告では，複素共役な特性乗数をもつ固定点に対して，偏角を指定してその位置，動径，パラメータ値を数値計算する方法を述べる．また，その応用として，Neimark-Sacker分岐パラメータ値を求める計算方法を提案する